Method and apparatus for ct image reconstruction

ABSTRACT

A method and apparatus for CT image reconstruction may include selecting, by a unit, projection data of the same height on a curve having a curvature approximate to that of the scanning circular orbit, implementing, by a unit, a weighting processing on the selected projection data, filtering, by a unit, the weighting processed projection data along a horizontal direction, implementing, by a unit, three-dimensional back projection on the filtered projection data along the direction of ray. The method and apparatus can effectively eliminate cone beam artifact under a large cone angle.

FIELD OF THE INVENTION

The present invention relates to technical fields of Computed Tomography(CT) imaging, and more particularly to a method and apparatus for CTimage reconstruction.

BACKGROUND INFORMATION

Since Hounsfield invented the first CT machine in 1972, CT technologyhas brought revolutionary influence to medical diagnosis and industrialnon-destructive detecting. CT has become an important detecting meansfor industries of medical treatment, biology, aeronautics, astronautics,national defense, etc. With the improvement of the technology, CTscanning modes and imaging methodologies have been continually improved,and three-dimensional cone-beam CT has become the mainstream of researchand application. X-ray cone-beam CT has been broadly applied in fieldsof clinical medicine, security inspection, non-destructive detecting,etc. In particular, because the cone-beam CT system based on circularorbit scanning is comparatively simple with respect to mechanics,electronic control, and so forth, and is easy for engineeringrealization, it is very broadly applied in clinical medicine, securityinspection, and industrial non-destructive detecting. In circular-orbitcone-beam reconstruction methods, the most broadly applied method is FDKmethod proposed by Feldkamp et al. (Feldkamp L. A., L. C. Davis, and J.W. Kress, Practical cone-beam algorithm, Journal of the Optical Societyof America, 1984, (1): 612-619).

The FDK method can be deemed as an approximate expansion to the fan-beamFBP (Filtered Backprojection) method under three-dimensional condition.The FDK method includes the following steps: initially implementingweighting processing on projection data; then implementingone-dimensional filtering on the projection data of different projectionangles in the horizontal direction; and finally implementingthree-dimensional back projection along the direction reverse to theX-ray to obtain a last three-dimensional reconstructed image of theobject.

Thus, it can be seen that the reconstructed voxel values of the FDKmethod are based on the sum of the contribution of radiation passingthrough the voxel in the projection angle range of 360 degrees.Accordingly, the circular-orbit cone-beam FDK method, as an approximatemethod, has the characteristics where the mathematical formula is simpleand the method computation is fast, and it is easy for engineeringrealization. Moreover, when the cone angle is comparatively small(generally within ±5°), it can achieve a very good reconstructioneffect, and it is therefore broadly appreciated in practical engineeringapplication.

However, the FDK method also has certain problems. Because the circularorbit scanning per se does not satisfy the condition of datacompleteness for cone-beam precise reconstruction, there exists theproblem of Radon data loss. Thus, when the cone angle of the cone beamincreases, the resulting image reconstructed by the FDK method willinclude serious cone beam artifact, and the FDK reconstruction valuewill decrease fast in a plane far away from the scanning orbit, suchthat the method is greatly limited in application in a CT imaging systemhaving a flat panel detector.

In order to improve the quality of image reconstruction of acircular-orbit FDK method under a large cone angle, based on the FDKmethod, a plurality of improved FDK methods are proposed, including, forexample, P-FDK (parallel FDK), T-FDK (tent-FDK), HT-FDK (hybridtent-FDK), EFDK (extended FDK), etc. In these improved FDK methods,because P-FDK and T-FDK are simple and can be easily implemented inengineering, they are comparatively broadly applied, and are thereforedescribed briefly below in greater detail.

The P-FDK method is to obtain parallel fan-beam projection data byrebinning cone-beam projection data, and then reconstructing athree-dimensional image of an object through a Filtered BackprojectionMethod. As shown in FIG. 1( a), which is a schematic diagram of a scanby a circular-orbit cone-beam CT system using a flat panel detector,S(f) indicates the location of the X-ray source on the circular orbit, βindicates the angular sampling location of projection on the circularorbit, P(β,a,b) indicates the projection data on the flat paneldetector, (a,b) is a rectangular coordinate system defined on the flatpanel detector for indicating the location coordinates of each X-rayprojected on the detector, R is the radius of the circular orbit. FIG.1( b) shows a rebinning of cone-beam projection into fan-beam projectionof parallel beams using the P-FDK method. The black solid dots representthe X-ray source, on a central virtual detector, in regard of therebinned projection data. Because the distances from the parallel fanbeams to the virtual detector under the same angle are different, a rowof projection on the original flat panel detector are not horizontal onthe corresponding virtual detector anymore. Instead, they are on a curveconvex along the central row of the virtual detector. Taking the flatpanel detector as an example, the data rebinning formula of P-FDK is asfollows:

$\begin{matrix}{{{P^{P\text{-}{FDK}}\left( {\theta,t,b} \right)} = {P\left( {{\theta - {\arcsin \; \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},b} \right)}},} & (1)\end{matrix}$

wherein θ indicates the sampling location of the rebinned projectiondata in angular direction after rebinning by P-FDK, (t,b) is therectangular coordinate system on the central virtual detector after therebinning, indicating the location coordinates of each X-ray on thevirtual detector after the rebinning. The subsequent derivationprocesses of the present invention also takes the flat panel detector asan example, other types of detectors, such as, for example, acylindrical surface detector, can be obtained by making correspondingchanges based on the flat panel detector, details of the changes notbeing further described herein.

The P-FDK method differs from the FDK method only in that the P-FDKmethod includes rebinning into parallel fan-beams such that the processof calculating weighting coefficients is omitted during back projection,while the method is not different from the FDK method in image quality.

The T-FDK method proposed by Grass et al. in 2000 provides animprovement. T-FDK provides for rebinning for a second time in avertical fan-beam plane, in addition to rebinning the cone-beamprojection into parallel fan beams. That is, T-FDK provides forrebinning projection data in both the horizontal and verticaldirections, ultimately causing the difference of T-FDK from P-FDK to bein that the direction of filtering the projection data according toT-FDK is along the horizontal direction of the central virtual detector,which is as shown in FIG. 1( c), rather than along the convex curvedirection. The data rebinning formula of T-FDK method is as follows:

$\begin{matrix}{{{P^{T\text{-}{FDK}}\left( {\theta,t,s} \right)} = {P\left( {{\theta - {\arcsin \; \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{{sR}^{2}}{R^{2} - t^{2}}} \right)}},} & (2)\end{matrix}$

where θ indicates the sampling location of the rebinned projection datain angular direction after rebinning according to T-FDK, and (t,s) isthe rectangular coordinate system on the central virtual detector afterthe rebinning according to T-FDK, indicating the location coordinates ofeach X-ray on the virtual detector after the rebinning.

T-FDK, compared to FDK, in one respect, is similar to P-FDK in that itprovides for rebinning into parallel fan beams such that the weightingcoefficient during back projection is omitted, and thus is moreefficient with respect to calculation. Meanwhile, because the filteringof projection data according to the T-FDK method is implemented alongthe horizontal direction of the central virtual detector, it reducescone beam artifact induced by increase of cone angle and improves imagereconstruction quality, such that it is possible to realize accuratethree-dimensional imaging of a large object using a large-area flatpanel detector through circular-orbit scanning. Besides, there isanother thought of improving the FDK method which uses conjugate rays ina circular-orbit scanning projection, different back projectionweighting coefficients being selected for conjugate projection so as toimprove the quality of the reconstructed image, and a comparatively goodeffect being also achieved.

With the gradual popularization of the flat panel detector, there aremore and more new types of cone-beam CT systems that use a large-areaflat panel detector, and the requirement for a large-cone-anglecircular-orbit cone-beam CT image reconstruction method is even greater.Taking the dental cone-beam CT apparatus which is comparatively broadlyapplied in dental disease diagnosis at present as an example,three-dimensional dental CT imaging of a plurality of manufacturers atpresent use a flat panel detector of 20 cm×25 cm, the distance from theX-ray source to the detector is 70 cm, and the size of the cone anglecorresponding to the circular orbit scanning is ±8.13°. Because doctorsmake disease diagnosis mainly dependent on CT value of image, i.e.,pixel value of the reconstructed image, the requirement forreconstruction value of image from the medical-use CT is very high. Suchlarge-cone-angle circular-orbit scanning goes far beyond the scope inwhich reconstruction can be done according to the FDK method, and thereconstruction result according to the T-FDK method is alsounsatisfactory. Moreover, with the rapid development of the detectortechnology, detectors having an even larger-area flat panel have beenapplied in clinical use. For example, flat panel detectors of differentsizes of 30 cm×40 cm, 43 cm×43 cm, etc. have been applied in clinical DR(Digital Radiography System). These detectors of even larger area cangreatly improve the effective detection area of cone-beam X-ray, enlargefield of view of imaging, and most importantly, can reduce or eveneliminate the problem where the CT value is unable to provide anaccurate reconstruction due to truncation of cone-beam projection data.Therefore, a large-area flat-panel detector can be very broadly appliedin current and future three-dimensional CT image apparatuses. However,with increase of the area of the flat panel detector, the cone angle ofthe three-dimensional CT system correspondingly increases, resulting inthe difficult problem of how to eliminate serious cone beam artifactunder a large cone angle.

SUMMARY OF THE INVENTION

A main technical problem to be solved by the invention is to provide amethod and apparatus for CT image reconstruction that can eliminateserious cone beam artifact under a large cone angle.

In order to solve the above-mentioned problem, the technical solution ofthe method for CT image reconstruction of the present invention includesthe steps of: selecting projection data of the same height on a curvehaving a curvature approximate to that of a scanning circular orbit;implementing weighting processing on the selected projection data;filtering the weighting processed projection data along a horizontaldirection; and implementing three-dimensional back projection on thefiltered projection data along the direction of a ray.

In an example embodiment, the step of selecting projection data of thesame height on a curve having a curvature approximate to that of thescanning circular orbit includes selecting projection data according tothe following formula:

${P^{C\text{-}{FDK}}\left( {\theta,t,c} \right)} = {P\left( {{\theta - {\arcsin \; \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{c \cdot R^{2}}{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}} \right)}$

where P^(C-FDK) (θ,t,c) indicates the selected projection data; θindicates the projection direction; t indicates the distance betweenparallel fan beams; c indicates the angular sampling interval in thedirection of Z axis; and R indicates the radius of the circular orbit.

In an example embodiment, the step of implementing weighting processingon the selected projection data includes processing the selectedprojection data according to the following formula:

${{\overset{\sim}{P}}^{C\text{-}{FDK}}\left( {\theta,t,c} \right)} = {\frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5R^{2}} - {4t^{2}} - {4R\sqrt{R^{2} - t^{2}}} + c^{2}}} \cdot {P^{C\text{-}{FDK}}\left( {\theta,t,c} \right)}}$

where, {tilde over (P)}^(C-FDK) (θ,t,c) indicates the weightingprocessed projection data.

In an example embodiment, the step of filtering the weighting processedprojection data along the horizontal direction includes filteringaccording to the following formula:

$\begin{matrix}{{g^{C\text{-}{FDK}}\left( {\theta,t,c} \right)} = {{{\overset{\sim}{P}}^{C\text{-}{FDK}}\left( {\theta,t,c} \right)} \otimes {h(t)}}} \\{= {\int_{- c_{0}}^{c_{0}}{{{{\overset{\sim}{P}}^{C\text{-}{FDK}}\left( {\theta,t,c} \right)} \cdot {h\left( {t - t^{\prime}} \right)}}{t^{\prime}}}}}\end{matrix}$

where g^(C-FDK) (θ,t,c) indicates the filtered projection data;

indicates the convolution; and h(t) indicates the filtering function.

In an example embodiment, the step of implementing three-dimensionalback projection on the filtered projection data along the direction ofthe ray includes implementing three-dimensional back projectionaccording to the following formula:

f^(C-FDK)(x, y, z) = ∫₀^(2π)g^(C-FDK)(θ, t(x, y, θ), c(x, y, z, θ))θ

where f^(C-FDK) (x,y,z) indicates the reconstructed image in thedirection of the X axis, Y axis and Z axis.

Correspondingly, an apparatus for CT image reconstruction according toan example embodiment of the present invention includes: a rebinningunit configured for selecting projection data of the same height on acurve having a curvature approximate to that of the scanning circularorbit; a weighting unit configured for implementing weighting processingon the selected projection data; a filtering unit configured forfiltering the weighting processed projection data along the horizontaldirection; and a back projection unit configured for implementingthree-dimensional back projection on the filtered projection data alongthe direction of the ray.

In an example embodiment, the rebinning unit is configured to selectprojection data according to the following formula:

${P^{C - {FDK}}\left( {\theta,t,c} \right)} = {P\left( {{\theta - {\arcsin \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{c \cdot R^{2}}{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}} \right)}$

where P^(C-FDK) (θ,t,c) indicates the selected projection data; θindicates the projection direction; t indicates the distance betweenparallel fan beams; c indicates the angular sampling interval in thedirection of Z axis; and R indicates the radius of the circular orbit.

In an example embodiment, the weighting unit is configured to processthe selected projection data according to the following formula:

${{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} = {\frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5\; R^{2}} - {4\; t^{2}} - {4\; R\sqrt{R^{2} - t^{2}}} + c^{2}}} \cdot {P^{C - {FDK}}\left( {\theta,t,c} \right)}}$

where, {tilde over (P)}^(C-FDK) (θ,t,c) indicates the weightingprocessed projection data.

In an example embodiment, the filtering unit is configured to implementone-dimensional ramp filtering according to the following formula:

$\begin{matrix}{{g^{C - {FDK}}\left( {\theta,t,c} \right)} = {{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \otimes {h(t)}}} \\{= {\int_{- c_{0}}^{c_{0}}{{{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \cdot {h\left( {t - t^{\prime}} \right)}}\ {t^{\prime}}}}}\end{matrix}$

where g^(C-FDK) (θ,c,t) indicates the filtered projection data;

indicates the convolution; and h(t) indicates the filtering function.

In an example embodiment, the back projection unit is configured toimplement three-dimensional back projection according to the followingformula:

f^(C − FDK)(x, y, z) = ∫₀^(2π)g^(C − FDK)(θ, t(x, y, θ), c(x, y, z, θ)) θ

where f^(C-FDK) (x,y,z) indicates the reconstructed image in thedirection of X axis, Y axis and Z axis.

Compared to the prior art, a beneficial effect of the method andapparatus for CT image reconstruction of the present invention includesthe following. The rebinning of projection data of the present inventionselects projection data of the same height on a curve having a curvatureapproximate to that of the scanning circular orbit, providing samplingon a curve concave to the central row along the virtual central detectorin the fan-beam plane parallel to the Z-axis, such that the numericalvalue accuracy of the reconstruction method under a large cone angle isgreatly improved, and cone angle artifact due to a large cone angle iseffectively inhibited.

In addition, the present invention is efficient in implementation andhas strong stability.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1( a) is a schematic diagram of conventional flat-panel-detectorcircular-orbit cone-beam CT scanning.

FIG. 1( b) is a schematic diagram of a state after rebinning projectiondata obtained by FIG. 1( a) using a conventional P-FDK method.

FIG. 1( c) is a schematic diagram a state after rebinning projectiondata obtained by FIG. 1( a) using a conventional T-FDK method.

FIG. 1( d) is a schematic diagram of a state after rebinning projectiondata obtained by FIG. 1( a) using a method for CT image reconstruction,according to an example embodiment of the present invention;

FIG. 2 is a top view showing the comparative features of the methodscorresponding to FIGS. 1( a), 1(b), 1(c), and 1(d) along the rotationaxis, i.e., Z axis.

FIG. 3 is a side view of the vertical section of FIG. 2 passing throughSC.

FIG. 4 is a flowchart of a method for CT image reconstruction, accordingto an example embodiment of the present invention.

FIG. 5( a) illustrates projection data acquired by circular-orbit CTscanning, which is a sinusoidal chart composed of data of the centrallayer of a flat panel detector under individual angles. according to anexample embodiment of the present invention???

FIG. 5( b) is a sinusoidal chart composed of data of the central layerof a virtual detector under individual angles after project data isrebinned, according to an example embodiment of the C-FDK method of thepresent invention.

FIG. 5( c) is a schematic diagram of projection data obtained afterimplementing weighting processing on the projection data shown in FIG.5( b), according to an example embodiment of the present invention.

FIG. 6( a) is a schematic diagram of an accurate image of athree-dimensional Shepp-logan head model in a vertical section.according to an example embodiment of the present invention???

FIG. 6( b) is a schematic diagram of a reconstruction result obtainedusing the FDK method.

FIG. 6( c) is a schematic diagram of a reconstruction result obtainedusing the T-FDK method.

FIG. 6( d) is a schematic diagram of a reconstruction result obtainedusing the C-FDK method, according to an example embodiment of thepresent invention.

FIG. 7 is a schematic diagram of a reconstruction result after selectinga section line along the vertical center of sections of FIGS. 6( a),6(b), 6(c), and 6(d).

FIG. 8 is a schematic diagram of an apparatus for CT imagereconstruction, according to an example embodiment of the presentinvention.

DETAILED DESCRIPTION

In order to understand the disclosed content more thoroughly, exampleembodiments are described below with reference to the figures. Whileexample embodiments of the invention are described in detail below, theinvention is not limited to the following embodiments.

As shown in FIG. 4, a method for CT image reconstruction according to anexample embodiment of the present invention includes the steps of: 1)selecting projection data of the same height on a curve having acurvature approximate to that of the scanning circular orbit; 2)implementing weighting processing on the selected projection data; 3)filtering the weighting processed projection data along the horizontaldirection; and 4) implementing three-dimensional back projection on thefiltered projection data along the direction of a ray. These steps maybe performed in a rebinning mode of the method for CT imagereconstruction of the present invention, and a three-dimensional CTimage of the scanned object can thereby be obtained.

For the curvature approximate to that of the scanning orbit, supposingthat the curvature of the scanning circular orbit is 1/R, theapproximate average curvature range may be, for example, 1/2R˜2R. Forthe filtering, any filtering method available to those skilled in theart may be employed. For example, the filter kernel may be a mostfundamental ramp filter kernel, and also may be a filter kernel obtainedfrom the standard ramp filtering that has been implemented withsmoothing processing in the frequency domain, for example, the commonlyused S-L filter kernel, etc. (The S-L filter kernel is that proposed byL. A. Shepp and B. F. Logan in 1974).

The technical solution of the present invention is described with a flatpanel detector as an example in the present description. Of course, thepresent invention may be applied to other panel detectors such as, forexample, a cylindrical detector.

As shown in FIG. 1( a), firstly, the scanning path of the cone-beamX-ray source in a plane may be defined as a circle {right arrow over(S)}(β)=R(cos β, sin β), where R is the radius of the circular orbit,and β indicates the angle parameter corresponding to the point of theray source point. In FIG. 1( a), O is the origin of coordinates and thecenter of the circular orbit, i.e., the rotation center. P(β,a,b)indicates projection data acquired on a flat panel detector 1 afterX-ray irradiates the scanned object, where (a,b) indicates horizontaland vertical coordinates of the location of a certain projection pointon the two-dimensional flat panel detector 1.

According to the characteristics of the P-FDK method, after rebinningcone-beam projection data, the method selects projection data on aconvex curve on a central virtual detector 2, as shown in FIG. 1( b).Referring to FIG. 2, which is a top view from top to bottom along therotation axis, i.e., Z axis, the panel detectors turn into a straightline or a curve, and the solid dot S on the circular orbit indicates theX-ray source. In FIG. 2, the process of rebinning data according toP-FDK method may be regarded as combining projection data of the sameheight on the curve in FIG. 2 that pass through OP, and then, data ofP-FDK, that is, the result of formula (1), is obtained, where the OPcurve is a curve defined by rebinning formula (1) of P-FDK. The processof rebinning data according to the T-FDK method may be regarded ascombining projection data of the same height on the straight line inFIG. 2 that pass through OO′, then, data of T-FDK, that is, the resultof formula (2), is obtained, where the OO′ curve is a curve defined byrebinning formula (2) of T-FDK.

The process of data rebinning of a method for CT image reconstructionaccording to an example embodiment of the present invention (referred toherein as C-FDK) may be regarded as combining projection data of thesame height on the curve in FIG. 2 that pass through OC. The OC curve isa segment of an arc with R as the radius. That is, it is a segment of anarc line having the same curvature as the scanning circular orbit, andthe center of the curvature of the OC curve is on a straight line thatpasses through point O and is parallel to SC. Of course, the curvatureof the OC curve may be slightly different than that of the scanningcircular orbit. For example, it may be within the range of 50% to 200%of the curvature of the circular orbit. That is, projection data withinthis range can also achieve the effect approximate to that of thedescribed example embodiment of the present invention.

The rebinning of projection data by the C-FDK method, according to anexample embodiment of the present invention is accomplished in the twodirections of horizontal and vertical of the central virtual detector.For example, it may be the same as the P-FDK and T-FDK methods in thehorizontal direction, and in the vertical direction it may be as shownin FIG. 3 for the rebinned data:

$\begin{matrix}{{c = {{\frac{SC}{SP} \cdot b} = {{\frac{{2\sqrt{R^{2} - t^{2}}} - R}{R^{2}/\sqrt{R^{2} - t^{2}}} \cdot b} = {\frac{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}{R^{2}} \cdot b}}}},} & (3)\end{matrix}$

where c indicates the coordinate of the central virtual detector in thevertical direction after the rebinning by the C-FDK method, S indicatesthe location of the intersection point at which the X-ray intersects theorbit, the X-ray has an angle θ, and the distance from the X-ray to thecentral ray is t. Thus, SC indicates the distance from the point of theX-ray source located at S to curve OC along said X-ray. SP indicates thedistance from the point of the X-ray source located at S to curve OPalong said X-ray.

FIG. 3 is a vertical section passing through SC in FIG. 2, where theline with an angle γ to the axis on which SC is present is the X-ray.According to the definition of the OC arc line in FIG. 2, the lengths ofSC, SP in FIG. 3 both can be obtained through calculation of spatial andgeometric relations, the specific lengths thereof being as follows:

SO′=√{square root over (R ² −t ²)}  (4)

SP=R ²/√{square root over (R ² −t ²)}  (5)

SC=2√{square root over (R ² −t ²)}−R  (6)

Thus, the formula of rebinning the circular-orbit cone-beam CTprojection data in an example embodiment of the C-FDK method of thepresent invention is as follows:

$\begin{matrix}{{P^{C - {FDK}}\left( {\theta,t,c} \right)} = {P\left( {{\theta - {\arcsin \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{c \cdot R^{2}}{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}} \right)}} & (7)\end{matrix}$

where θ indicates the projection direction, t indicates the distancebetween parallel fan beams, and c indicates the angular samplinginterval in the direction of Z axis.

After rebinning data by the C-FDK method using the above formula (7),the next step is implementing weighting processing on the projectiondata. The weighting coefficient may be cos γ, where γ is the anglebetween each X-ray and the projection line on the central planeprojected by said X-ray along the direction of Z axis, as shown in FIG.3, according to the spatial and geometric relations:

$\begin{matrix}{{\cos \; \gamma} = {\frac{SC}{\sqrt{{SC}^{2} + c^{2}}} = \frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5\; R^{2}} - {4\; t^{2}} - {4\; R\sqrt{R^{2} - t^{2}}} + c^{2}}}}} & (8)\end{matrix}$

That is, the formula of implementing weighting processing on theprojection is:

$\begin{matrix}{{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} = {\frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5\; R^{2}} - {4\; t^{2}} - {4\; R\sqrt{R^{2} - t^{2}}} + c^{2}}} \cdot {P^{C - {FDK}}\left( {\theta,t,c} \right)}}} & (9)\end{matrix}$

Then, one-dimensional ramp filtering may be implemented on theabove-processed projection data along the horizontal direction:

$\begin{matrix}\begin{matrix}{{g^{C - {FDK}}\left( {\theta,t,c} \right)} = {{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \otimes {h(t)}}} \\{= {\int_{- c_{0}}^{c_{0}}{{{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \cdot {h\left( {t - t^{\prime}} \right)}}\ {t^{\prime}}}}}\end{matrix} & (10)\end{matrix}$

In the formula above,

indicates convolution, and h(t) is a filtering function, for which aRamp filter, for example, may be employed.

Lastly, implementing three-dimensional back projection on the filteredprojection data along the direction of ray, whereby a three-dimensionalCT image of the scanned object can be reconstructed, the back projectionformula, in an example embodiment, is as follows:

$\begin{matrix}{{f^{C - {FDK}}\left( {x,y,z} \right)} = {\int_{0}^{2\pi}{{g^{C - {FDK}}\left( {\theta,{t\left( {x,y,\theta} \right)},{c\left( {x,y,z,\theta} \right)}} \right)}\ {\theta}}}} & (11)\end{matrix}$

where the calculation formula of the projection location on the detectoris as follows:

$\begin{matrix}{{t\left( {x,y,\theta} \right)} = {{y\; \cos \; \theta} - {x\; \sin \; \theta}}} & (12) \\{{c\left( {x,y,z,\theta} \right)} = \frac{z \cdot \left( {{2\sqrt{R^{2} - t^{2}}} - R} \right)}{\sqrt{R^{2} - t^{2}} + {x\; \cos \; \theta} + {y\; \sin \; \theta}}} & (13)\end{matrix}$

FIG. 5( a) shows projection data acquired by circular-orbit CT scanning,which is a sinusoidal chart composed of data of the central layer of theflat panel detector under individual angles, while FIG. 5( b) is asinusoidal chart composed of data of the central layer of a virtualdetector under individual angles after projection data is rebinned bythe C-FDK method according to an example embodiment of the presentinvention. FIG. 5( c) is projection data obtained after implementingweighting processing on the projection data shown in FIG. 5( b). Then,one-dimensional ramp filtering may be implemented on the projection datathat has been implemented with weighting processing, and, lastly, backprojection is implemented, whereby a CT image of the scanned object canbe obtained, as shown in FIG. 6( d).

Correspondingly, as shown in FIG. 8, an apparatus for CT imagereconstruction according to an example embodiment of the presentinvention includes: a rebinning unit 1 configured to select projectiondata of the same height on a curve having a curvature approximate tothat of the scanning circular orbit; a weighting unit 2 configured toimplement weighting processing on the selected projection data; afiltering unit 3 configured to filter the weighting processed projectiondata along the horizontal direction; and back projection unit 4configured to implement three-dimensional back projection on thefiltered projection data along the direction of ray.

Preferably, the rebinning unit 1 is configured to select projection dataaccording to the following formula:

${P^{C - {FDK}}\left( {\theta,t,c} \right)} = {P\left( {{\theta - {\arcsin \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{c \cdot R^{2}}{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}} \right)}$

where P^(C-FDK) (θ,t,c) indicates the selected projection data; θindicates the projection direction, t indicates the distance betweenparallel fan beams, c indicates the angular sampling interval in thedirection of Z axis, and R indicates the radius of the circular orbit.

Preferably, said weighting unit 2 is configured to process the selectedprojection data according to the following formula:

${{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} = {\frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5\; R^{2}} - {4\; t^{2}} - {4\; R\sqrt{R^{2} - t^{2}}} + c^{2}}} \cdot {P^{C - {FDK}}\left( {\theta,t,c} \right)}}$

where {tilde over (P)}^(C-FDK) (θ,t,c) indicates the weighting processedprojection data.

Preferably, said filtering unit 3 is configured to implement filteringaccording to the following formula:

$\begin{matrix}{{g^{C - {FDK}}\left( {\theta,t,c} \right)} = {{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \otimes {h(t)}}} \\{= {\int_{- c_{0}}^{c_{0}}{{{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \cdot {h\left( {t - t^{\prime}} \right)}}\ {t^{\prime}}}}}\end{matrix}$

where g^(C-FDK) (θ,t,c) indicates the filtered projection data,

indicates the convolution, and h(t) indicates the filtering function.

Preferably, said back projection unit 4 is configured to implementthree-dimensional back projection according to the following formula:

f^(C − FDK)(x, y, z) = ∫₀^(2π)g^(C − FDK)(θ, t(x, y, θ), c(x, y, z, θ)) θ

where f^(C-FDK) (x,y,z) indicates the reconstructed image in thedirection of X axis, Y axis, and Z axis.

Because the technical features of the apparatus for CT imagereconstruction of the present invention correspond to the technicalfeatures described with respect to the method for CT reconstruction ofthe present invention, therefore, the technical solution of theapparatus for CT image reconstruction of the present invention will notbe described in detail.

The inventor of the present application has performed a numericalsimulative experiment using the three-dimensional Shepp-Logan headmodel, and has made an experimental comparison with the FDK and T-FDKmethods to verify the technical solutions of the present invention. Inthe numerical simulative experiment, the three-dimensional head modelwas limited within a 1 mm globe, the center of the model was therotation center of CT scanning, the distance from the X-ray source tothe rotation center was 4 mm, the distance from the X-ray source to thedetector was 8 mm, the size of the flat panel detector was 4 mm×4 mm,the number of the detecting units was 256×256, 360 cone-beam projectionswere acquired uniformly in angle within the range of 360 degrees, andthen a three-dimensional CT image was reconstructed. According to thegeometric definition above, it can be calculated that in the experiment,the largest cone angle of the cone-beam X-ray is 14 degrees. That is,the X-ray cone angle range of this experiment is ±14°. With thecircular-orbit cone-beam projection data under said above-describedscanning condition, three-dimensional CT images were reconstructedrespectively employing the FDK method, T-FDK method, and C-FDK method ofthe present invention.

FIG. 6( a) is an accurate image of the Shepp-logan head model in avertical section, where said vertical section is selected parallel tothe x-z plane and where the three-dimensional head model is at thelocation y=−0.25. FIG. 6( b) is a reconstruction result obtained byemploying the FDK method. FIG. 6( c) is a reconstruction result obtainedby employing the T-FDK method. FIG. 6( d) is a reconstruction resultobtained by employing the C-FDK method according to an exampleembodiment of the present invention.

It can be seen from each of the reconstructed images that the C-FDKmethod of the present invention can nicely reconstruct an image of thethree-dimensional head model under the condition of ±14°, and itovercomes the difficult problem of cone beam artifact which isubiquitous in large-cone-angle circular-orbit CT reconstruction in theexisting methods, and nicely solves the problem of large-cone-anglecircular-orbit CT image reconstruction.

In order to more accurately analyze the accuracy of the numerical valueof the reconstruction result, FIG. 7 selects a section line taken alongthe vertical center of the section of FIG. 6, where the abscissaindicates the length coordinate of the points on the section line, theordinate indicates the linear attenuation coefficient of the points onthe section line, the long dashed line indicates the reconstructionresult of the FDK method, the dash-dotted line indicates thereconstruction result of the T-FDK method, the short dashed lineindicates the reconstruction result of the C-FDK method of the presentinvention, and the solid line indicates the accurate numerical value ofthe head model. It can be seen more clearly from the section line thatwhen the cone angle increases, the reconstruction numerical values ofthe existing FDK and T-FDK methods show a rapidly decreasing trend, andgo farther and farther from the accurate value of the model, while theC-FDK method of the present invention still can comparatively accuratelyreconstruct the accurate value of the original model even if it iswithin the cone angle range of ±14°, and fundamentally solves thedifficult problem of large-cone-angle circular-orbit CT imagereconstruction.

A core concept of the C-FDK method, according to example embodiments ofthe present invention, lies in the implementation of filtering onprojection data for a virtual detector that is concave along themidline, and, in the derivation processes above, example embodiments ofthe present invention provide for rebinning X-ray and the projection ofthe same height on the OC curve in FIG. 2 into C-FDK data P^(C-FDK)(θ,t,c) of the same height c. While the OC curve has been described as asegment of an arc having the same curvature as the circular scanningorbit, variations of such OC curve may be used instead. For example, avariation within a small range may be made around the curvature in otherexample embodiments of the present invention.

Those skilled in the art can appreciate from the foregoing descriptionthat the present invention may be implemented in a variety of forms,that the various embodiments may be implemented alone or in combination,and that the above described example embodiments are not used forlimiting the present invention. Therefore, while the embodiments of thepresent invention have been described in connection with particularexamples thereof, the true scope of the embodiments of the presentinvention should not be so limited since other modifications will becomeapparent to the skilled practitioner upon a study of the drawings,specification, and following claims. Many duplicate and alternativesolutions, including modifications, additions, permutations, andvariations, will be apparent to those skilled in the art in light of thedisclosed content of the present application and should fall within theprotection scope of the present invention. Those skilled in the art mayimplement various variations, modifications, and equivalentsubstitutions to the described features of the invention withoutdeparting from the true spirit and scope of the invention. Suchvariations, modifications, and equivalent substitutions are intended tofall within the spirit and scope defined by the following claims.

1-10. (canceled)
 11. A method for CT image reconstruction, comprising:selecting projection data of a same height on a curve whose curvature isapproximate to that of a scanning circular orbit; implementing, by acomputer processor, weighting processing on the selected projectiondata; filtering, by the processor, the weighting processed projectiondata along a horizontal direction; and implementing, by the processor,three-dimensional back projection on the filtered projection data alonga direction of a ray.
 12. The method for CT image reconstructionaccording to claim 11, wherein: the selecting of the projection dataincludes selecting projection data according to the formula${{P^{C - {FDK}}\left( {\theta,t,c} \right)} = {P\left( {{\theta - {\arcsin \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{c \cdot R^{2}}{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}} \right)}};$P^(C-FDK) (θ,t,c) indicates the selected projection data; θ indicates aprojection direction; t indicates a distance between parallel fan beams;c indicates an angular sampling interval in a Z axis direction; and Rindicates a radius of the circular orbit.
 13. The method for CT imagereconstruction according to claim 12, wherein: the implementing of theweighting processing includes processing the selected projection dataaccording to the formula${{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} = {\frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5\; R^{2}} - {4\; t^{2}} - {4\; R\sqrt{R^{2} - t^{2}}} + c^{2}}} \cdot {P^{C - {FDK}}\left( {\theta,t,c} \right)}}};$and {tilde over (P)}^(C-FDK) (θ,t,c) indicates the weighting processedprojection data.
 14. The method for CT image reconstruction according toclaim 13, wherein the filtering includes filtering according to theformula $\begin{matrix}{{g^{C - {FDK}}\left( {\theta,t,c} \right)} = {{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \otimes {h(t)}}} \\{{= {\int_{- c_{0}}^{c_{0}}{{{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \cdot {h\left( {t - t^{\prime}} \right)}}\ {t^{\prime}}}}};}\end{matrix}$ g^(C-FDK) (θ,t,c) indicates the filtered projection data;

indicates a convolution; and h(t) indicates a filtering function. 15.The method for CT image reconstruction according to claim 14, whereinthe implementing of the three-dimensional back projection includesimplementing three-dimensional back projection according to the formulaf^(C − FDK)(x, y, z) = ∫₀^(2π)g^(C − FDK)(θ, t(x, y, θ), c(x, y, z, θ)) θ;and f^(C-FDK) (x,y,z) indicates the reconstructed image in a directionof X axis, Y axis, and Z axis.
 16. An apparatus for CT imagereconstruction, comprising: a rebinning unit configured to selectprojection data of a same height on a curve whose curvature isapproximate to that of a scanning circular orbit; a weighting unitconfigured to implement weighting processing on the selected projectiondata; a filtering unit configured to filter the weighting processedprojection data; and a back projection unit configured to implementthree-dimensional back projection on the filtered projection data alonga direction of a ray.
 17. The apparatus for CT image reconstructionaccording to claim 16, wherein the rebinning unit is configured toselect projection data according to the formula${{P^{C - {FDK}}\left( {\theta,t,c} \right)} = {P\left( {{\theta - {\arcsin \frac{t}{R}}},\frac{tR}{\sqrt{R^{2} - t^{2}}},\frac{c \cdot R^{2}}{{2\left( {R^{2} - t^{2}} \right)} - {R\sqrt{R^{2} - t^{2}}}}} \right)}};$P^(C-FDK) (θ,t,c) indicates the selected projection data; θ indicates aprojection direction; t indicates a distance between parallel fan beams;c indicates an angular sampling interval in a Z axis direction; and Rindicates a radius of the circular orbit.
 18. The apparatus for CT imagereconstruction according to claim 17, wherein the weighting unit isconfigured to process the selected projection data according to theformula${{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} = {\frac{{2\sqrt{R^{2} - t^{2}}} - R}{\sqrt{{5\; R^{2}} - {4\; t^{2}} - {4\; R\sqrt{R^{2} - t^{2}}} + c^{2}}} \cdot {P^{C - {FDK}}\left( {\theta,t,c} \right)}}};$and {tilde over (P)}^(C-FDK) (θ,t,c) indicates the weighting processedprojection data.
 19. The apparatus for CT image reconstruction accordingto claim 18, wherein the filtering unit is configured to implementone-dimensional ramp filtering according to the g^(C-FDK) (θ,t,c)={tildeover (P)}^(C-FDK) (θ,t,c)

h(t) formula;$= {\int_{- c_{0}}^{c_{0}}{{{{\overset{\sim}{P}}^{C - {FDK}}\left( {\theta,t,c} \right)} \cdot {h\left( {t - t^{\prime}} \right)}}\ {t^{\prime}}}}$g^(C-FDK) (θ,t,c) indicates the filtered projection data;

indicates a convolution; and h(t) indicates a filtering function. 20.The apparatus for CT image reconstruction according to claim 19, whereinthe back projection unit is configured to implement three-dimensionalback projection according to the formulaf^(C − FDK)(x, y, z) = ∫₀^(2π)g^(C − FDK)(θ, t(x, y, θ), c(x, y, z, θ)) θ;and f^(C-FDK) (x,y,z) indicates the reconstructed image in a directionof X axis, Y axis, and Z axis.